What Is The Factorization Of The Trinomial Below? − X 2 + X + 42 -x^2 + X + 42 − X 2 + X + 42 A. ( − X + 6 ) ( X + 7 (-x + 6)(x + 7 ( − X + 6 ) ( X + 7 ] B. ( X + 6 ) ( X − 7 (x + 6)(x - 7 ( X + 6 ) ( X − 7 ] C. − 1 ( X − 7 ) ( X + 6 -1(x - 7)(x + 6 − 1 ( X − 7 ) ( X + 6 ] D. − 1 ( X + 7 ) ( X + 6 -1(x + 7)(x + 6 − 1 ( X + 7 ) ( X + 6 ]

Understanding Trinomial Factorization

Trinomial factorization is a process in algebra where we express a trinomial, which is a polynomial with three terms, as a product of two binomials. This process is essential in solving quadratic equations and simplifying expressions. In this article, we will focus on finding the factorization of the given trinomial, $-x^2 + x + 42$.

The Importance of Trinomial Factorization

Trinomial factorization has numerous applications in mathematics and other fields. It is used to solve quadratic equations, which are equations of the form $ax^2 + bx + c = 0$. By factoring the trinomial, we can find the roots of the quadratic equation, which are the values of $x$ that satisfy the equation. This is particularly useful in physics, engineering, and economics, where quadratic equations are used to model real-world problems.

The Factorization Process

To factorize a trinomial, we need to find two binomials whose product is equal to the trinomial. The general form of a trinomial is $ax^2 + bx + c$. We can factorize it by finding two numbers whose product is $ac$ and whose sum is $b$. These numbers are called the factors of the trinomial.

Finding the Factors

To find the factors of the trinomial $-x^2 + x + 42$, we need to find two numbers whose product is $-42$ and whose sum is $1$. We can start by listing the factors of $-42$, which are:

• $1$ and $-42$
• $-1$ and $42$
• $2$ and $-21$
• $-2$ and $21$
• $3$ and $-14$
• $-3$ and $14$
• $6$ and $-7$
• $-6$ and $7$

Identifying the Correct Factors

We need to find the correct factors that satisfy the condition that their sum is $1$. By examining the list of factors, we can see that the correct factors are $6$ and $-7$. Therefore, the factorization of the trinomial $-x^2 + x + 42$ is:

$-x^2 + x + 42 = (x + 6)(x - 7)$

Conclusion

In this article, we have discussed the factorization of the trinomial $-x^2 + x + 42$. We have explained the importance of trinomial factorization and the process of finding the factors of a trinomial. By identifying the correct factors, we have found that the factorization of the trinomial is $(x + 6)(x - 7)$. This result is consistent with option B in the given multiple-choice question.

Final Answer

The final answer is:

(x + 6)(x - 7)

Q: What is trinomial factorization?

A: Trinomial factorization is a process in algebra where we express a trinomial, which is a polynomial with three terms, as a product of two binomials.

Q: Why is trinomial factorization important?

A: Trinomial factorization has numerous applications in mathematics and other fields. It is used to solve quadratic equations, which are equations of the form $ax^2 + bx + c = 0$. By factoring the trinomial, we can find the roots of the quadratic equation, which are the values of $x$ that satisfy the equation.

Q: How do I factorize a trinomial?

A: To factorize a trinomial, we need to find two binomials whose product is equal to the trinomial. The general form of a trinomial is $ax^2 + bx + c$. We can factorize it by finding two numbers whose product is $ac$ and whose sum is $b$. These numbers are called the factors of the trinomial.

Q: What are the steps to factorize a trinomial?

A: The steps to factorize a trinomial are:

1. Identify the trinomial and its coefficients.
2. Find the factors of the product of the coefficients.
3. Identify the correct factors that satisfy the condition that their sum is the coefficient of the middle term.
4. Write the factorization as a product of two binomials.

Q: How do I find the factors of a trinomial?

A: To find the factors of a trinomial, we need to find two numbers whose product is the product of the coefficients and whose sum is the coefficient of the middle term. We can start by listing the factors of the product of the coefficients and then identifying the correct factors that satisfy the condition.

Q: What are some common mistakes to avoid when factorizing a trinomial?

A: Some common mistakes to avoid when factorizing a trinomial include:

• Not identifying the correct factors.
• Not checking if the factors satisfy the condition that their sum is the coefficient of the middle term.
• Not writing the factorization as a product of two binomials.

Q: Can you give an example of trinomial factorization?

A: Yes, let's consider the trinomial $-x^2 + x + 42$. We can factorize it by finding two numbers whose product is $-42$ and whose sum is $1$. The correct factors are $6$ and $-7$, so the factorization of the trinomial is:

$-x^2 + x + 42 = (x + 6)(x - 7)$

Q: How do I check if the factorization is correct?

A: To check if the factorization is correct, we can multiply the two binomials and see if we get the original trinomial. If we do, then the factorization is correct.

Q: What are some real-world applications of trinomial factorization?

A: Trinomial factorization has numerous real-world applications, including:

• Solving quadratic equations in physics and engineering.
• Modeling population growth and decline in biology.
• Analyzing financial data in economics.

Q: Can you give some tips for mastering trinomial factorization?

A: Yes, here are some tips for mastering trinomial factorization:

• Practice, practice, practice! The more you practice, the more comfortable you will become with the process.
• Start with simple trinomials and work your way up to more complex ones.
• Use online resources and video tutorials to help you understand the process.
• Join a study group or find a study partner to help you stay motivated and get help when you need it.